We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1. $2^{O(\sqrt{k}\log k)} \cdot n^{O(1)}$ time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a $2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)}$ time algorithm for Edge Multiway Cut and a $2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)}$ time algorithm for Vertex Multiway Cut, where $r$ is the number of terminals to be separated; 3. a $2^{O((r+\sqrt{k})\log^4 (rk))} \cdot n^{O(1)}$ time algorithm for Edge Multicut and a $2^{O((\sqrt{rk}+r) \log (rk))} \cdot n^{O(1)}$ time algorithm for Vertex Multicut, where $r$ is the number of terminal pairs to be separated; 4. a $2^{O(\sqrt{k} \log g \log^4 k)} \cdot n^{O(1)}$ time algorithm for Group Feedback Edge Set and a $2^{O(g \sqrt{k}\log(gk))} \cdot n^{O(1)}$ time algorithm for Group Feedback Vertex Set, where $g$ is the size of the group. 5. In addition, our approach also gives $n^{O(\sqrt{k})}$ time algorithms for all above problems with the exception of $n^{O(r+\sqrt{k})}$ time for Edge/Vertex Multicut and $(ng)^{O(\sqrt{k})}$ time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an $h$-almost-embeddable graph for any fixed constant $h$. In particular we show the following. Let $G$ be an $h$-almost-embeddable graph for a constant $h$. Then for every $p\in\mathbb{N}$, there exist disjoint sets $Z_1,\dots,Z_p \subseteq V(G)$ such that for every $i \in \{1,\dots,p\}$ and every $Z'\subseteq Z_i$, the treewidth of $G/(Z_i\backslash Z')$ is $O(p+|Z'|)$. Here $G/(Z_i\backslash Z')$ is the graph obtained from $G$ by contracting edges with both endpoints in $Z_i \backslash Z'$.